Question

Which book is this question from If the horizontal range of projectile be (a) and the maximum height attained by it is (b) then prove that the velocity of projection is

Answer 1

Step-by-step explanation:

Let the body be projected with a velocity u making an angle ß with the horizontal direction. We can resolve u into a horizontal component u' = u Cos ß and a vertical component u''= u Sin ß.

Let us find the time of flight of the projectile. It is the time taken by the projectile to hit the ground at the level from which the projectile was fired.

The projectile will rise with initial velocity u Sin ß till such time as its velocity reduces to zero under the opposing action of gravity. Using the relation:v =u + a t

In our case the initial upwards velocity is u'= u Sin ß, v=0, and a= -g,

The time taken 't' by the projectile to rise to the maximum height = (0 - u Sinß)/(- g)= u Sinß/g

Time taken by the projectile to descend from the maximum height= Time taken by the projectile to ascend to the highest point. So,

The time of flight T of the projectile= 2× time taken to ascend to the highest point = 2 u Sin ß/g–––––——(1)

The range of the projectile= Horizontal component of projection velocity × Time of flight= (u Cos ß) ×(2u Sinß/g) = (u²/g) 2 Sin ß Ços ß = a ( we are given range is a) ——————————————————————(2)

To find the maximum height attained by the projectile, we shall use the relation: v² - u² = 2 a h.

In our case u =u'= u Sin ß, v=0, a = -g, and h= b (given).

0² - ( u Sin ß)² = - 2 g b or b= (u² Sin² ß)/2g ————(3)

We have determined that

a =( u²/g) 2 Sin ß Cos ß;

==> a² = ( u² Cos²ß/g²)(4 u² Sin²ß)————————-(4)

From (3) we know that 4 u² Sin²ß = 8 g b —————(5)

Substituting from (5) in (4) we get,

a² = (u² Cos²ß/g²) × 8 g b; ==> u² Cos² ß = g a²/8 b —-(6)

u² Sin²ß = 2 g b (from eq. 3)———————————(7)

Adding (6) and (7) we get

u² = 2g b + g a²/8b= 2 g b + 2 g a²/16 b= 2g ( b + a²/16b)

Or

u = [ 2 g ( b + a²/16 b)]^½

The desired relation is proved.

Answer 2

Answer:

YES THE ABOVE ANSWER IS FULLY CORRECT